Theorem lubfun 18485

Description: The LUB is a function. (Contributed by NM, 9-Sep-2018.)

Hypothesis

Ref	Expression
lubfun.u	⊢ 𝑈 = (lub‘𝐾)

Assertion

Ref	Expression
lubfun	⊢ Fun 𝑈

Proof of Theorem lubfun

Dummy variables 𝑥 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funmpt 6566	. . . 4 ⊢ Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
2		funres 6570	. . . 4 ⊢ (Fun (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) → Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
3	1, 2	ax-mp 5	. . 3 ⊢ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})
4		eqid 2760	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2760	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
6		lubfun.u	. . . . 5 ⊢ 𝑈 = (lub‘𝐾)
7		biid 264	. . . . 5 ⊢ ((∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
8		id 23	. . . . 5 ⊢ (𝐾 ∈ V → 𝐾 ∈ V)
9	4, 5, 6, 7, 8	lubfval 18483	. . . 4 ⊢ (𝐾 ∈ V → 𝑈 = ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))}))
10	9	funeqd 6549	. . 3 ⊢ (𝐾 ∈ V → (Fun 𝑈 ↔ Fun ((𝑠 ∈ 𝒫 (Base‘𝐾) ↦ (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))) ↾ {𝑠 ∣ ∃!𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑠 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))})))
11	3, 10	mpbiri 261	. 2 ⊢ (𝐾 ∈ V → Fun 𝑈)
12		fun0 6593	. . 3 ⊢ Fun ∅
13		fvprc 6865	. . . . 5 ⊢ (¬ 𝐾 ∈ V → (lub‘𝐾) = ∅)
14	6, 13	eqtrid 2807	. . . 4 ⊢ (¬ 𝐾 ∈ V → 𝑈 = ∅)
15	14	funeqd 6549	. . 3 ⊢ (¬ 𝐾 ∈ V → (Fun 𝑈 ↔ Fun ∅))
16	12, 15	mpbiri 261	. 2 ⊢ (¬ 𝐾 ∈ V → Fun 𝑈)
17	11, 16	pm2.61i 184	1 ⊢ Fun 𝑈

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃!wreu 3363  Vcvv 3450  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185   ↾ cres 5649  Fun wfun 6521  ‘cfv 6527  ℩crio 7364  Basecbs 17348  lecple 17396  lubclub 18444

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-lub 18479

This theorem is used by:  joinfval  18506  joinfval2  18507